Question #25248

Show that the following are equivalent:
(A) If I is a nil ideal in any ring R, then Mn(I) is nil for any n.
(B)' If I is a nil ideal in any ring, then M2(I) is nil.

Expert's answer

That (A) implies (B) is obvious,since it is case n=2.For, if (B) holds, then for any nilideal I, m=2^{t} , M_{m} (I) is isomorphic to M_{2}(M_{m/2}(I)) is also nil, by induction on t. Since any n is bounded by 2^{t }forsome t, we have (A).

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